Product Cocycles and the Approximate Transitivity

Some criteria of the approximate transitivity in the terms of Mackey actions and product cocycles are proved. The Mackey action constructed by an amenable type II or III transformation group G and a 1-cocycle ρ x α, where ρ is the Radon-Nikodym cocycle while α is an arbitrary 1-cocycle with values in a locally compact separable group A, is approximately transitive (AT) if and only if the pair (G,(ρ, α)) is weakly equivalent to a product odometer supplied with a product cocycle. Besides, in the case when the given AT action from the very beginning was a range of a type II action and a nontransient cocycle, then this cocycle turns out to be cohomologous to a θ-product cocycle. An example is constructed that shows that it is necessary to consider the double Mackey actions since they can not be reduced to the single ones